Lecture 10 Graph Algorithmselearning.kocw.net/KOCW/document/2015/sungkyunkwan/... · Testing for articulation vertices or bridges •Brute force •Be sure to add that vertex/edge

1 SWE2004: Principles in Programming | Spring 2014 | Euiseong Seo ([email protected])

Lecture 10 Graph Algorithms

Euiseong Seo

([email protected])


Graph Theory

Study of the properties of graph structures

• It provides us with a language with which to talk about graphs

Keys to solving problems

• Identifying the fundamental graph theoretic notion underlying the situation

• Using classical algorithms to solve the resulting problem


Degrees

Number of edges connected to a vertex

For undirected graphs

• Sum of all degrees = 2 * edges

For directed graph

• Sum of in-degree = sum of out-degree


Connectivity

A graph is connected if there is an undirected path

between every pair of vertices

Existence of a spanning tree -> connectivity

• BFS or DFS connected component algorithms to find connected components

Articulation vertex

• A single vertex whose deletion disconnects the graph

Biconnected graphs

• Any graphs without an articulation vertex

Bridge

• A single edge whose deletion disconnects the graph


Connectivity

Testing for articulation vertices or bridges

• Brute force

• Be sure to add that vertex/edge back before doing the next deletion!

Strongly connected components

• For directed graphs only

• Partitions the graph into chunks such that there are directed paths between all pairs of vertices within a given chunk

• Road networks should be strongly connected


Cycles in Graphs

All non-tree connected graphs contain cycles

Eulerian cycle

• A tour which visits every edge of the graph exactly once

• Condition for a undirected graph to have an Eulerian cycle?

• Find an Eulerian cycle by building it once cycle at a time – Finding a back edge using DFS

– Deleting the edges on this cycle leaves each vertex with even degree


Cycles in Graphs

Hamiltonian cycles

• A tour which visits every vertex of the graph exactly once

• Traveling salesman problem = the shortest Hamiltonian cycle on a weighted graph

• No efficient algorithm to find one – Backtracking!


Spanning Tree

Given a graph G = (V, E) and tree T = (V, E) – E E

– for all (u, v) in E u, v V

– for all connected graph, there exists a spanning tree

A spanning tree can be constructed using DFS or BFS

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Minimum Spanning Tree

A spanning tree whose sum of edge weights is minimal

Importance of MSTs

• When to need connect a set of points by the smallest amount of roadway, wire or pipe

Prim’s and Kruskal’s algorithms


Example of MST: Prim’s Algorithm


1. Vertex D has been chosen as a starting point ① Vertices A, B, E, F are connected to D through a single edge. ② A is the nearest to D and thus chosen as the 2nd vertex along

with the edge AD


2. The next vertex chosen is the vertex nearest to either D or A. So the vertex F is chosen along with the edge DF


3. same as 2, Vertex B is chosen.


4. among C, E, G, E is chosen.


5. among C, G, C is chosen.


6. G is the only remaining vertex. E is chosen.


7. The finally obtained minimum spanning tree

the total weight is 39


Applications of Prim’s

Maximum spanning trees

Minimum product spanning trees

Minimum bottleneck spanning tree


Shortest Path

For unweighted graphs

• BFS suffices

For weighted graphs

• Dijkstra’s algorithm

Dijkstra’s algorithm is very similar to Prim’s


Dijkstra’s Algorithm

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All Pairs Shortest Path

When you want to know the center vertex for a new business

• Dijksta’s?

We need another one

• Floyd’s all pairs shortest path algorithm


All Pairs Shortest Path

Floyd’s algorithm

• W(i,j,k) = min(W(i,j,k-1), W(i,k,k-1)+W(k,j,k-1))


Transitive Closure

Sometimes we are interested in which vertices are

reachable from a given node

The Blackmail problem


Network Flow

Any edge-weighted graph can be thought of as a network of pipes

• Weight of edge (i, j) measures the capacity of the pipe

Network flow problem

• Given source and sink vertices of a graph

• The maximum amount of flow which can be sent from source to sink while respecting the maximum capacities of each pipe


Bipartite Graphs

A graph G is bipartite or two-colorable if the vertices can be divided into two sets

A matching in a graph G = (V, E) is a subset of edges E’ ⊂ E such that no two edges in E’ share a vertex

• Job-worker pairs

• Marriage graphs

The largest possible bipartite matching can be found using network flow

Documents

Lecture 10 Graph Algorithmselearning.kocw.net/KOCW/document/2015/sungkyunkwan/... · Testing for articulation vertices or bridges •Brute force •Be sure to add that vertex/edge